An element $x$ of a semigroup $S$ is called regular provided that there exists $y\in S$ such that $xyx=x$. $S$ is called regular if all its elements are regular. Let $S$ be a monoid with identity element $1$. An element $a\in S$ is called invertible if there exists $b\in S$ such that $ab=ba=1$. The set of all inverse elements of the monoid $S$ is denoted by $S^{\star}$.
Are there examples of monoids that are non-regular and have more than one invertible element, particularly related to transformation semigroups?
Best Answer
The integers under multiplication has two invertible elements, yet it only has three regular elements.
The dyadic rationals (the rational numbers whose denominator is a natural power of $2$) under multiplication has infinitely many invertible elements (any integral power of $2$), yet any dyadic rational with an odd prime factor in its numerator is non-regular.